LMFDB - Embedded newform 513.2.a.c.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [513,2,Mod(1,513)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("513.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(513, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 513 = 3^{3} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	513.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$4.09632562369$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	513.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.73205 q^{2} +1.00000 q^{4} +3.46410 q^{5} -4.00000 q^{7} +1.73205 q^{8} -6.00000 q^{10} -5.19615 q^{11} -4.00000 q^{13} +6.92820 q^{14} -5.00000 q^{16} +3.46410 q^{17} +1.00000 q^{19} +3.46410 q^{20} +9.00000 q^{22} -3.46410 q^{23} +7.00000 q^{25} +6.92820 q^{26} -4.00000 q^{28} +1.73205 q^{29} -7.00000 q^{31} +5.19615 q^{32} -6.00000 q^{34} -13.8564 q^{35} -10.0000 q^{37} -1.73205 q^{38} +6.00000 q^{40} -1.73205 q^{41} +8.00000 q^{43} -5.19615 q^{44} +6.00000 q^{46} -8.66025 q^{47} +9.00000 q^{49} -12.1244 q^{50} -4.00000 q^{52} -8.66025 q^{53} -18.0000 q^{55} -6.92820 q^{56} -3.00000 q^{58} +3.46410 q^{59} -1.00000 q^{61} +12.1244 q^{62} +1.00000 q^{64} -13.8564 q^{65} -13.0000 q^{67} +3.46410 q^{68} +24.0000 q^{70} +3.46410 q^{71} -7.00000 q^{73} +17.3205 q^{74} +1.00000 q^{76} +20.7846 q^{77} -1.00000 q^{79} -17.3205 q^{80} +3.00000 q^{82} +5.19615 q^{83} +12.0000 q^{85} -13.8564 q^{86} -9.00000 q^{88} +1.73205 q^{89} +16.0000 q^{91} -3.46410 q^{92} +15.0000 q^{94} +3.46410 q^{95} +2.00000 q^{97} -15.5885 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4} - 8 q^{7} - 12 q^{10} - 8 q^{13} - 10 q^{16} + 2 q^{19} + 18 q^{22} + 14 q^{25} - 8 q^{28} - 14 q^{31} - 12 q^{34} - 20 q^{37} + 12 q^{40} + 16 q^{43} + 12 q^{46} + 18 q^{49} - 8 q^{52} - 36 q^{55}+ \cdots + 4 q^{97}+O(q^{100}) $ 2 * q + 2 * q^4 - 8 * q^7 - 12 * q^10 - 8 * q^13 - 10 * q^16 + 2 * q^19 + 18 * q^22 + 14 * q^25 - 8 * q^28 - 14 * q^31 - 12 * q^34 - 20 * q^37 + 12 * q^40 + 16 * q^43 + 12 * q^46 + 18 * q^49 - 8 * q^52 - 36 * q^55 - 6 * q^58 - 2 * q^61 + 2 * q^64 - 26 * q^67 + 48 * q^70 - 14 * q^73 + 2 * q^76 - 2 * q^79 + 6 * q^82 + 24 * q^85 - 18 * q^88 + 32 * q^91 + 30 * q^94 + 4 * q^97

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.73205	−1.22474	−0.612372	−	0.790569i	$-0.709785\pi$
$2$	−1.73205	−1.22474	−0.612372	+	0.790569i	$0.709785\pi$
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	3.46410	1.54919	0.774597	−	0.632456i	$-0.217953\pi$
$5$	3.46410	1.54919	0.774597	+	0.632456i	$0.217953\pi$
$6$	0	0
$7$	−4.00000	−1.51186	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	1.73205	0.612372
$9$	0	0
$10$	−6.00000	−1.89737
$11$	−5.19615	−1.56670	−0.783349	−	0.621582i	$-0.786490\pi$
$11$	−5.19615	−1.56670	−0.783349	+	0.621582i	$0.786490\pi$
$12$	0	0
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	6.92820	1.85164
$15$	0	0
$16$	−5.00000	−1.25000
$17$	3.46410	0.840168	0.420084	−	0.907485i	$-0.362001\pi$
$17$	3.46410	0.840168	0.420084	+	0.907485i	$0.362001\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	3.46410	0.774597
$21$	0	0
$22$	9.00000	1.91881
$23$	−3.46410	−0.722315	−0.361158	−	0.932505i	$-0.617618\pi$
$23$	−3.46410	−0.722315	−0.361158	+	0.932505i	$0.617618\pi$
$24$	0	0
$25$	7.00000	1.40000
$26$	6.92820	1.35873
$27$	0	0
$28$	−4.00000	−0.755929
$29$	1.73205	0.321634	0.160817	−	0.986984i	$-0.448587\pi$
$29$	1.73205	0.321634	0.160817	+	0.986984i	$0.448587\pi$
$30$	0	0
$31$	−7.00000	−1.25724	−0.628619	−	0.777714i	$-0.716379\pi$
$31$	−7.00000	−1.25724	−0.628619	+	0.777714i	$0.716379\pi$
$32$	5.19615	0.918559
$33$	0	0
$34$	−6.00000	−1.02899
$35$	−13.8564	−2.34216
$36$	0	0
$37$	−10.0000	−1.64399	−0.821995	−	0.569495i	$-0.807139\pi$
$37$	−10.0000	−1.64399	−0.821995	+	0.569495i	$0.807139\pi$
$38$	−1.73205	−0.280976
$39$	0	0
$40$	6.00000	0.948683
$41$	−1.73205	−0.270501	−0.135250	−	0.990811i	$-0.543184\pi$
$41$	−1.73205	−0.270501	−0.135250	+	0.990811i	$0.543184\pi$
$42$	0	0
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	−5.19615	−0.783349
$45$	0	0
$46$	6.00000	0.884652
$47$	−8.66025	−1.26323	−0.631614	−	0.775283i	$-0.717607\pi$
$47$	−8.66025	−1.26323	−0.631614	+	0.775283i	$0.717607\pi$
$48$	0	0
$49$	9.00000	1.28571
$50$	−12.1244	−1.71464
$51$	0	0
$52$	−4.00000	−0.554700
$53$	−8.66025	−1.18958	−0.594789	−	0.803882i	$-0.702764\pi$
$53$	−8.66025	−1.18958	−0.594789	+	0.803882i	$0.702764\pi$
$54$	0	0
$55$	−18.0000	−2.42712
$56$	−6.92820	−0.925820
$57$	0	0
$58$	−3.00000	−0.393919
$59$	3.46410	0.450988	0.225494	−	0.974245i	$-0.427600\pi$
$59$	3.46410	0.450988	0.225494	+	0.974245i	$0.427600\pi$
$60$	0	0
$61$	−1.00000	−0.128037	−0.0640184	−	0.997949i	$-0.520392\pi$
$61$	−1.00000	−0.128037	−0.0640184	+	0.997949i	$0.520392\pi$
$62$	12.1244	1.53979
$63$	0	0
$64$	1.00000	0.125000
$65$	−13.8564	−1.71868
$66$	0	0
$67$	−13.0000	−1.58820	−0.794101	−	0.607785i	$-0.792058\pi$
$67$	−13.0000	−1.58820	−0.794101	+	0.607785i	$0.792058\pi$
$68$	3.46410	0.420084
$69$	0	0
$70$	24.0000	2.86855
$71$	3.46410	0.411113	0.205557	−	0.978645i	$-0.434100\pi$
$71$	3.46410	0.411113	0.205557	+	0.978645i	$0.434100\pi$
$72$	0	0
$73$	−7.00000	−0.819288	−0.409644	−	0.912245i	$-0.634347\pi$
$73$	−7.00000	−0.819288	−0.409644	+	0.912245i	$0.634347\pi$
$74$	17.3205	2.01347
$75$	0	0
$76$	1.00000	0.114708
$77$	20.7846	2.36863
$78$	0	0
$79$	−1.00000	−0.112509	−0.0562544	−	0.998416i	$-0.517916\pi$
$79$	−1.00000	−0.112509	−0.0562544	+	0.998416i	$0.517916\pi$
$80$	−17.3205	−1.93649
$81$	0	0
$82$	3.00000	0.331295
$83$	5.19615	0.570352	0.285176	−	0.958475i	$-0.407948\pi$
$83$	5.19615	0.570352	0.285176	+	0.958475i	$0.407948\pi$
$84$	0	0
$85$	12.0000	1.30158
$86$	−13.8564	−1.49417
$87$	0	0
$88$	−9.00000	−0.959403
$89$	1.73205	0.183597	0.0917985	−	0.995778i	$-0.470738\pi$
$89$	1.73205	0.183597	0.0917985	+	0.995778i	$0.470738\pi$
$90$	0	0
$91$	16.0000	1.67726
$92$	−3.46410	−0.361158
$93$	0	0
$94$	15.0000	1.54713
$95$	3.46410	0.355409
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	−15.5885	−1.57467
$99$	0	0
$100$	7.00000	0.700000
$101$	17.3205	1.72345	0.861727	−	0.507371i	$-0.169383\pi$
$101$	17.3205	1.72345	0.861727	+	0.507371i	$0.169383\pi$
$102$	0	0
$103$	−16.0000	−1.57653	−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000	−1.57653	−0.788263	+	0.615338i	$0.789020\pi$
$104$	−6.92820	−0.679366
$105$	0	0
$106$	15.0000	1.45693
$107$	10.3923	1.00466	0.502331	−	0.864675i	$-0.332476\pi$
$107$	10.3923	1.00466	0.502331	+	0.864675i	$0.332476\pi$
$108$	0	0
$109$	−4.00000	−0.383131	−0.191565	−	0.981480i	$-0.561356\pi$
$109$	−4.00000	−0.383131	−0.191565	+	0.981480i	$0.561356\pi$
$110$	31.1769	2.97260
$111$	0	0
$112$	20.0000	1.88982
$113$	−5.19615	−0.488813	−0.244406	−	0.969673i	$-0.578593\pi$
$113$	−5.19615	−0.488813	−0.244406	+	0.969673i	$0.578593\pi$
$114$	0	0
$115$	−12.0000	−1.11901
$116$	1.73205	0.160817
$117$	0	0
$118$	−6.00000	−0.552345
$119$	−13.8564	−1.27021
$120$	0	0
$121$	16.0000	1.45455
$122$	1.73205	0.156813
$123$	0	0
$124$	−7.00000	−0.628619
$125$	6.92820	0.619677
$126$	0	0
$127$	−4.00000	−0.354943	−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000	−0.354943	−0.177471	+	0.984126i	$0.556792\pi$
$128$	−12.1244	−1.07165
$129$	0	0
$130$	24.0000	2.10494
$131$	3.46410	0.302660	0.151330	−	0.988483i	$-0.451644\pi$
$131$	3.46410	0.302660	0.151330	+	0.988483i	$0.451644\pi$
$132$	0	0
$133$	−4.00000	−0.346844
$134$	22.5167	1.94514
$135$	0	0
$136$	6.00000	0.514496
$137$	17.3205	1.47979	0.739895	−	0.672722i	$-0.234875\pi$
$137$	17.3205	1.47979	0.739895	+	0.672722i	$0.234875\pi$
$138$	0	0
$139$	14.0000	1.18746	0.593732	−	0.804663i	$-0.297654\pi$
$139$	14.0000	1.18746	0.593732	+	0.804663i	$0.297654\pi$
$140$	−13.8564	−1.17108
$141$	0	0
$142$	−6.00000	−0.503509
$143$	20.7846	1.73810
$144$	0	0
$145$	6.00000	0.498273
$146$	12.1244	1.00342
$147$	0	0
$148$	−10.0000	−0.821995
$149$	−17.3205	−1.41895	−0.709476	−	0.704730i	$-0.751068\pi$
$149$	−17.3205	−1.41895	−0.709476	+	0.704730i	$0.751068\pi$
$150$	0	0
$151$	17.0000	1.38344	0.691720	−	0.722166i	$-0.256853\pi$
$151$	17.0000	1.38344	0.691720	+	0.722166i	$0.256853\pi$
$152$	1.73205	0.140488
$153$	0	0
$154$	−36.0000	−2.90096
$155$	−24.2487	−1.94770
$156$	0	0
$157$	5.00000	0.399043	0.199522	−	0.979893i	$-0.436061\pi$
$157$	5.00000	0.399043	0.199522	+	0.979893i	$0.436061\pi$
$158$	1.73205	0.137795
$159$	0	0
$160$	18.0000	1.42302
$161$	13.8564	1.09204
$162$	0	0
$163$	−10.0000	−0.783260	−0.391630	−	0.920123i	$-0.628089\pi$
$163$	−10.0000	−0.783260	−0.391630	+	0.920123i	$0.628089\pi$
$164$	−1.73205	−0.135250
$165$	0	0
$166$	−9.00000	−0.698535
$167$	6.92820	0.536120	0.268060	−	0.963402i	$-0.413617\pi$
$167$	6.92820	0.536120	0.268060	+	0.963402i	$0.413617\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	−20.7846	−1.59411
$171$	0	0
$172$	8.00000	0.609994
$173$	−12.1244	−0.921798	−0.460899	−	0.887453i	$-0.652473\pi$
$173$	−12.1244	−0.921798	−0.460899	+	0.887453i	$0.652473\pi$
$174$	0	0
$175$	−28.0000	−2.11660
$176$	25.9808	1.95837
$177$	0	0
$178$	−3.00000	−0.224860
$179$	−10.3923	−0.776757	−0.388379	−	0.921500i	$-0.626965\pi$
$179$	−10.3923	−0.776757	−0.388379	+	0.921500i	$0.626965\pi$
$180$	0	0
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	−27.7128	−2.05421
$183$	0	0
$184$	−6.00000	−0.442326
$185$	−34.6410	−2.54686
$186$	0	0
$187$	−18.0000	−1.31629
$188$	−8.66025	−0.631614
$189$	0	0
$190$	−6.00000	−0.435286
$191$	15.5885	1.12794	0.563971	−	0.825795i	$-0.309273\pi$
$191$	15.5885	1.12794	0.563971	+	0.825795i	$0.309273\pi$
$192$	0	0
$193$	−4.00000	−0.287926	−0.143963	−	0.989583i	$-0.545985\pi$
$193$	−4.00000	−0.287926	−0.143963	+	0.989583i	$0.545985\pi$
$194$	−3.46410	−0.248708
$195$	0	0
$196$	9.00000	0.642857
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	0	0
$199$	20.0000	1.41776	0.708881	−	0.705328i	$-0.249200\pi$
$199$	20.0000	1.41776	0.708881	+	0.705328i	$0.249200\pi$
$200$	12.1244	0.857321
$201$	0	0
$202$	−30.0000	−2.11079
$203$	−6.92820	−0.486265
$204$	0	0
$205$	−6.00000	−0.419058
$206$	27.7128	1.93084
$207$	0	0
$208$	20.0000	1.38675
$209$	−5.19615	−0.359425
$210$	0	0
$211$	−13.0000	−0.894957	−0.447478	−	0.894295i	$-0.647678\pi$
$211$	−13.0000	−0.894957	−0.447478	+	0.894295i	$0.647678\pi$
$212$	−8.66025	−0.594789
$213$	0	0
$214$	−18.0000	−1.23045
$215$	27.7128	1.89000
$216$	0	0
$217$	28.0000	1.90076
$218$	6.92820	0.469237
$219$	0	0
$220$	−18.0000	−1.21356
$221$	−13.8564	−0.932083
$222$	0	0
$223$	−1.00000	−0.0669650	−0.0334825	−	0.999439i	$-0.510660\pi$
$223$	−1.00000	−0.0669650	−0.0334825	+	0.999439i	$0.510660\pi$
$224$	−20.7846	−1.38873
$225$	0	0
$226$	9.00000	0.598671
$227$	10.3923	0.689761	0.344881	−	0.938647i	$-0.387919\pi$
$227$	10.3923	0.689761	0.344881	+	0.938647i	$0.387919\pi$
$228$	0	0
$229$	−7.00000	−0.462573	−0.231287	−	0.972886i	$-0.574293\pi$
$229$	−7.00000	−0.462573	−0.231287	+	0.972886i	$0.574293\pi$
$230$	20.7846	1.37050
$231$	0	0
$232$	3.00000	0.196960
$233$	3.46410	0.226941	0.113470	−	0.993541i	$-0.463803\pi$
$233$	3.46410	0.226941	0.113470	+	0.993541i	$0.463803\pi$
$234$	0	0
$235$	−30.0000	−1.95698
$236$	3.46410	0.225494
$237$	0	0
$238$	24.0000	1.55569
$239$	15.5885	1.00833	0.504167	−	0.863606i	$-0.331800\pi$
$239$	15.5885	1.00833	0.504167	+	0.863606i	$0.331800\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	−27.7128	−1.78145
$243$	0	0
$244$	−1.00000	−0.0640184
$245$	31.1769	1.99182
$246$	0	0
$247$	−4.00000	−0.254514
$248$	−12.1244	−0.769897
$249$	0	0
$250$	−12.0000	−0.758947
$251$	−3.46410	−0.218652	−0.109326	−	0.994006i	$-0.534869\pi$
$251$	−3.46410	−0.218652	−0.109326	+	0.994006i	$0.534869\pi$
$252$	0	0
$253$	18.0000	1.13165
$254$	6.92820	0.434714
$255$	0	0
$256$	19.0000	1.18750
$257$	6.92820	0.432169	0.216085	−	0.976375i	$-0.430671\pi$
$257$	6.92820	0.432169	0.216085	+	0.976375i	$0.430671\pi$
$258$	0	0
$259$	40.0000	2.48548
$260$	−13.8564	−0.859338
$261$	0	0
$262$	−6.00000	−0.370681
$263$	−12.1244	−0.747620	−0.373810	−	0.927505i	$-0.621949\pi$
$263$	−12.1244	−0.747620	−0.373810	+	0.927505i	$0.621949\pi$
$264$	0	0
$265$	−30.0000	−1.84289
$266$	6.92820	0.424795
$267$	0	0
$268$	−13.0000	−0.794101
$269$	−27.7128	−1.68968	−0.844840	−	0.535019i	$-0.820304\pi$
$269$	−27.7128	−1.68968	−0.844840	+	0.535019i	$0.820304\pi$
$270$	0	0
$271$	14.0000	0.850439	0.425220	−	0.905090i	$-0.360197\pi$
$271$	14.0000	0.850439	0.425220	+	0.905090i	$0.360197\pi$
$272$	−17.3205	−1.05021
$273$	0	0
$274$	−30.0000	−1.81237
$275$	−36.3731	−2.19338
$276$	0	0
$277$	−19.0000	−1.14160	−0.570800	−	0.821089i	$-0.693367\pi$
$277$	−19.0000	−1.14160	−0.570800	+	0.821089i	$0.693367\pi$
$278$	−24.2487	−1.45434
$279$	0	0
$280$	−24.0000	−1.43427
$281$	−13.8564	−0.826604	−0.413302	−	0.910594i	$-0.635625\pi$
$281$	−13.8564	−0.826604	−0.413302	+	0.910594i	$0.635625\pi$
$282$	0	0
$283$	8.00000	0.475551	0.237775	−	0.971320i	$-0.423582\pi$
$283$	8.00000	0.475551	0.237775	+	0.971320i	$0.423582\pi$
$284$	3.46410	0.205557
$285$	0	0
$286$	−36.0000	−2.12872
$287$	6.92820	0.408959
$288$	0	0
$289$	−5.00000	−0.294118
$290$	−10.3923	−0.610257
$291$	0	0
$292$	−7.00000	−0.409644
$293$	5.19615	0.303562	0.151781	−	0.988414i	$-0.451499\pi$
$293$	5.19615	0.303562	0.151781	+	0.988414i	$0.451499\pi$
$294$	0	0
$295$	12.0000	0.698667
$296$	−17.3205	−1.00673
$297$	0	0
$298$	30.0000	1.73785
$299$	13.8564	0.801337
$300$	0	0
$301$	−32.0000	−1.84445
$302$	−29.4449	−1.69436
$303$	0	0
$304$	−5.00000	−0.286770
$305$	−3.46410	−0.198354
$306$	0	0
$307$	5.00000	0.285365	0.142683	−	0.989769i	$-0.454427\pi$
$307$	5.00000	0.285365	0.142683	+	0.989769i	$0.454427\pi$
$308$	20.7846	1.18431
$309$	0	0
$310$	42.0000	2.38544
$311$	25.9808	1.47323	0.736617	−	0.676310i	$-0.236422\pi$
$311$	25.9808	1.47323	0.736617	+	0.676310i	$0.236422\pi$
$312$	0	0
$313$	−7.00000	−0.395663	−0.197832	−	0.980236i	$-0.563390\pi$
$313$	−7.00000	−0.395663	−0.197832	+	0.980236i	$0.563390\pi$
$314$	−8.66025	−0.488726
$315$	0	0
$316$	−1.00000	−0.0562544
$317$	6.92820	0.389127	0.194563	−	0.980890i	$-0.437671\pi$
$317$	6.92820	0.389127	0.194563	+	0.980890i	$0.437671\pi$
$318$	0	0
$319$	−9.00000	−0.503903
$320$	3.46410	0.193649
$321$	0	0
$322$	−24.0000	−1.33747
$323$	3.46410	0.192748
$324$	0	0
$325$	−28.0000	−1.55316
$326$	17.3205	0.959294
$327$	0	0
$328$	−3.00000	−0.165647
$329$	34.6410	1.90982
$330$	0	0
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	5.19615	0.285176
$333$	0	0
$334$	−12.0000	−0.656611
$335$	−45.0333	−2.46043
$336$	0	0
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	−5.19615	−0.282633
$339$	0	0
$340$	12.0000	0.650791
$341$	36.3731	1.96971
$342$	0	0
$343$	−8.00000	−0.431959
$344$	13.8564	0.747087
$345$	0	0
$346$	21.0000	1.12897
$347$	3.46410	0.185963	0.0929814	−	0.995668i	$-0.470360\pi$
$347$	3.46410	0.185963	0.0929814	+	0.995668i	$0.470360\pi$
$348$	0	0
$349$	23.0000	1.23116	0.615581	−	0.788074i	$-0.288921\pi$
$349$	23.0000	1.23116	0.615581	+	0.788074i	$0.288921\pi$
$350$	48.4974	2.59230
$351$	0	0
$352$	−27.0000	−1.43910
$353$	−31.1769	−1.65938	−0.829690	−	0.558225i	$-0.811483\pi$
$353$	−31.1769	−1.65938	−0.829690	+	0.558225i	$0.811483\pi$
$354$	0	0
$355$	12.0000	0.636894
$356$	1.73205	0.0917985
$357$	0	0
$358$	18.0000	0.951330
$359$	−17.3205	−0.914141	−0.457071	−	0.889430i	$-0.651101\pi$
$359$	−17.3205	−0.914141	−0.457071	+	0.889430i	$0.651101\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−24.2487	−1.27448
$363$	0	0
$364$	16.0000	0.838628
$365$	−24.2487	−1.26924
$366$	0	0
$367$	−28.0000	−1.46159	−0.730794	−	0.682598i	$-0.760850\pi$
$367$	−28.0000	−1.46159	−0.730794	+	0.682598i	$0.760850\pi$
$368$	17.3205	0.902894
$369$	0	0
$370$	60.0000	3.11925
$371$	34.6410	1.79847
$372$	0	0
$373$	−4.00000	−0.207112	−0.103556	−	0.994624i	$-0.533022\pi$
$373$	−4.00000	−0.207112	−0.103556	+	0.994624i	$0.533022\pi$
$374$	31.1769	1.61212
$375$	0	0
$376$	−15.0000	−0.773566
$377$	−6.92820	−0.356821
$378$	0	0
$379$	−1.00000	−0.0513665	−0.0256833	−	0.999670i	$-0.508176\pi$
$379$	−1.00000	−0.0513665	−0.0256833	+	0.999670i	$0.508176\pi$
$380$	3.46410	0.177705
$381$	0	0
$382$	−27.0000	−1.38144
$383$	34.6410	1.77007	0.885037	−	0.465521i	$-0.154133\pi$
$383$	34.6410	1.77007	0.885037	+	0.465521i	$0.154133\pi$
$384$	0	0
$385$	72.0000	3.66946
$386$	6.92820	0.352636
$387$	0	0
$388$	2.00000	0.101535
$389$	6.92820	0.351274	0.175637	−	0.984455i	$-0.443802\pi$
$389$	6.92820	0.351274	0.175637	+	0.984455i	$0.443802\pi$
$390$	0	0
$391$	−12.0000	−0.606866
$392$	15.5885	0.787336
$393$	0	0
$394$	0	0
$395$	−3.46410	−0.174298
$396$	0	0
$397$	11.0000	0.552074	0.276037	−	0.961147i	$-0.410979\pi$
$397$	11.0000	0.552074	0.276037	+	0.961147i	$0.410979\pi$
$398$	−34.6410	−1.73640
$399$	0	0
$400$	−35.0000	−1.75000
$401$	13.8564	0.691956	0.345978	−	0.938243i	$-0.387547\pi$
$401$	13.8564	0.691956	0.345978	+	0.938243i	$0.387547\pi$
$402$	0	0
$403$	28.0000	1.39478
$404$	17.3205	0.861727
$405$	0	0
$406$	12.0000	0.595550
$407$	51.9615	2.57564
$408$	0	0
$409$	−10.0000	−0.494468	−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000	−0.494468	−0.247234	+	0.968956i	$0.579522\pi$
$410$	10.3923	0.513239
$411$	0	0
$412$	−16.0000	−0.788263
$413$	−13.8564	−0.681829
$414$	0	0
$415$	18.0000	0.883585
$416$	−20.7846	−1.01905
$417$	0	0
$418$	9.00000	0.440204
$419$	15.5885	0.761546	0.380773	−	0.924669i	$-0.375658\pi$
$419$	15.5885	0.761546	0.380773	+	0.924669i	$0.375658\pi$
$420$	0	0
$421$	20.0000	0.974740	0.487370	−	0.873195i	$-0.337956\pi$
$421$	20.0000	0.974740	0.487370	+	0.873195i	$0.337956\pi$
$422$	22.5167	1.09609
$423$	0	0
$424$	−15.0000	−0.728464
$425$	24.2487	1.17624
$426$	0	0
$427$	4.00000	0.193574
$428$	10.3923	0.502331
$429$	0	0
$430$	−48.0000	−2.31477
$431$	−34.6410	−1.66860	−0.834300	−	0.551311i	$-0.814128\pi$
$431$	−34.6410	−1.66860	−0.834300	+	0.551311i	$0.814128\pi$
$432$	0	0
$433$	20.0000	0.961139	0.480569	−	0.876957i	$-0.340430\pi$
$433$	20.0000	0.961139	0.480569	+	0.876957i	$0.340430\pi$
$434$	−48.4974	−2.32795
$435$	0	0
$436$	−4.00000	−0.191565
$437$	−3.46410	−0.165710
$438$	0	0
$439$	−19.0000	−0.906821	−0.453410	−	0.891302i	$-0.649793\pi$
$439$	−19.0000	−0.906821	−0.453410	+	0.891302i	$0.649793\pi$
$440$	−31.1769	−1.48630
$441$	0	0
$442$	24.0000	1.14156
$443$	−39.8372	−1.89272	−0.946360	−	0.323113i	$-0.895271\pi$
$443$	−39.8372	−1.89272	−0.946360	+	0.323113i	$0.895271\pi$
$444$	0	0
$445$	6.00000	0.284427
$446$	1.73205	0.0820150
$447$	0	0
$448$	−4.00000	−0.188982
$449$	36.3731	1.71655	0.858276	−	0.513189i	$-0.171536\pi$
$449$	36.3731	1.71655	0.858276	+	0.513189i	$0.171536\pi$
$450$	0	0
$451$	9.00000	0.423793
$452$	−5.19615	−0.244406
$453$	0	0
$454$	−18.0000	−0.844782
$455$	55.4256	2.59839
$456$	0	0
$457$	23.0000	1.07589	0.537947	−	0.842978i	$-0.319200\pi$
$457$	23.0000	1.07589	0.537947	+	0.842978i	$0.319200\pi$
$458$	12.1244	0.566534
$459$	0	0
$460$	−12.0000	−0.559503
$461$	−38.1051	−1.77473	−0.887366	−	0.461065i	$-0.847467\pi$
$461$	−38.1051	−1.77473	−0.887366	+	0.461065i	$0.847467\pi$
$462$	0	0
$463$	−34.0000	−1.58011	−0.790057	−	0.613033i	$-0.789949\pi$
$463$	−34.0000	−1.58011	−0.790057	+	0.613033i	$0.789949\pi$
$464$	−8.66025	−0.402042
$465$	0	0
$466$	−6.00000	−0.277945
$467$	5.19615	0.240449	0.120225	−	0.992747i	$-0.461639\pi$
$467$	5.19615	0.240449	0.120225	+	0.992747i	$0.461639\pi$
$468$	0	0
$469$	52.0000	2.40114
$470$	51.9615	2.39681
$471$	0	0
$472$	6.00000	0.276172
$473$	−41.5692	−1.91135
$474$	0	0
$475$	7.00000	0.321182
$476$	−13.8564	−0.635107
$477$	0	0
$478$	−27.0000	−1.23495
$479$	−3.46410	−0.158279	−0.0791394	−	0.996864i	$-0.525217\pi$
$479$	−3.46410	−0.158279	−0.0791394	+	0.996864i	$0.525217\pi$
$480$	0	0
$481$	40.0000	1.82384
$482$	17.3205	0.788928
$483$	0	0
$484$	16.0000	0.727273
$485$	6.92820	0.314594
$486$	0	0
$487$	−43.0000	−1.94852	−0.974258	−	0.225436i	$-0.927619\pi$
$487$	−43.0000	−1.94852	−0.974258	+	0.225436i	$0.927619\pi$
$488$	−1.73205	−0.0784063
$489$	0	0
$490$	−54.0000	−2.43947
$491$	−12.1244	−0.547165	−0.273582	−	0.961849i	$-0.588209\pi$
$491$	−12.1244	−0.547165	−0.273582	+	0.961849i	$0.588209\pi$
$492$	0	0
$493$	6.00000	0.270226
$494$	6.92820	0.311715
$495$	0	0
$496$	35.0000	1.57155
$497$	−13.8564	−0.621545
$498$	0	0
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	6.92820	0.309839
$501$	0	0
$502$	6.00000	0.267793
$503$	1.73205	0.0772283	0.0386142	−	0.999254i	$-0.487706\pi$
$503$	1.73205	0.0772283	0.0386142	+	0.999254i	$0.487706\pi$
$504$	0	0
$505$	60.0000	2.66996
$506$	−31.1769	−1.38598
$507$	0	0
$508$	−4.00000	−0.177471
$509$	1.73205	0.0767718	0.0383859	−	0.999263i	$-0.487778\pi$
$509$	1.73205	0.0767718	0.0383859	+	0.999263i	$0.487778\pi$
$510$	0	0
$511$	28.0000	1.23865
$512$	−8.66025	−0.382733
$513$	0	0
$514$	−12.0000	−0.529297
$515$	−55.4256	−2.44234
$516$	0	0
$517$	45.0000	1.97910
$518$	−69.2820	−3.04408
$519$	0	0
$520$	−24.0000	−1.05247
$521$	−20.7846	−0.910590	−0.455295	−	0.890341i	$-0.650466\pi$
$521$	−20.7846	−0.910590	−0.455295	+	0.890341i	$0.650466\pi$
$522$	0	0
$523$	11.0000	0.480996	0.240498	−	0.970650i	$-0.422689\pi$
$523$	11.0000	0.480996	0.240498	+	0.970650i	$0.422689\pi$
$524$	3.46410	0.151330
$525$	0	0
$526$	21.0000	0.915644
$527$	−24.2487	−1.05629
$528$	0	0
$529$	−11.0000	−0.478261
$530$	51.9615	2.25706
$531$	0	0
$532$	−4.00000	−0.173422
$533$	6.92820	0.300094
$534$	0	0
$535$	36.0000	1.55642
$536$	−22.5167	−0.972572
$537$	0	0
$538$	48.0000	2.06943
$539$	−46.7654	−2.01433
$540$	0	0
$541$	−7.00000	−0.300954	−0.150477	−	0.988614i	$-0.548081\pi$
$541$	−7.00000	−0.300954	−0.150477	+	0.988614i	$0.548081\pi$
$542$	−24.2487	−1.04157
$543$	0	0
$544$	18.0000	0.771744
$545$	−13.8564	−0.593543
$546$	0	0
$547$	−28.0000	−1.19719	−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000	−1.19719	−0.598597	+	0.801050i	$0.704275\pi$
$548$	17.3205	0.739895
$549$	0	0
$550$	63.0000	2.68633
$551$	1.73205	0.0737878
$552$	0	0
$553$	4.00000	0.170097
$554$	32.9090	1.39817
$555$	0	0
$556$	14.0000	0.593732
$557$	−34.6410	−1.46779	−0.733893	−	0.679265i	$-0.762299\pi$
$557$	−34.6410	−1.46779	−0.733893	+	0.679265i	$0.762299\pi$
$558$	0	0
$559$	−32.0000	−1.35346
$560$	69.2820	2.92770
$561$	0	0
$562$	24.0000	1.01238
$563$	−20.7846	−0.875967	−0.437983	−	0.898983i	$-0.644307\pi$
$563$	−20.7846	−0.875967	−0.437983	+	0.898983i	$0.644307\pi$
$564$	0	0
$565$	−18.0000	−0.757266
$566$	−13.8564	−0.582428
$567$	0	0
$568$	6.00000	0.251754
$569$	−20.7846	−0.871336	−0.435668	−	0.900107i	$-0.643488\pi$
$569$	−20.7846	−0.871336	−0.435668	+	0.900107i	$0.643488\pi$
$570$	0	0
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	20.7846	0.869048
$573$	0	0
$574$	−12.0000	−0.500870
$575$	−24.2487	−1.01124
$576$	0	0
$577$	−34.0000	−1.41544	−0.707719	−	0.706494i	$-0.750276\pi$
$577$	−34.0000	−1.41544	−0.707719	+	0.706494i	$0.750276\pi$
$578$	8.66025	0.360219
$579$	0	0
$580$	6.00000	0.249136
$581$	−20.7846	−0.862291
$582$	0	0
$583$	45.0000	1.86371
$584$	−12.1244	−0.501709
$585$	0	0
$586$	−9.00000	−0.371787
$587$	45.0333	1.85872	0.929362	−	0.369170i	$-0.120358\pi$
$587$	45.0333	1.85872	0.929362	+	0.369170i	$0.120358\pi$
$588$	0	0
$589$	−7.00000	−0.288430
$590$	−20.7846	−0.855689
$591$	0	0
$592$	50.0000	2.05499
$593$	27.7128	1.13803	0.569014	−	0.822328i	$-0.307325\pi$
$593$	27.7128	1.13803	0.569014	+	0.822328i	$0.307325\pi$
$594$	0	0
$595$	−48.0000	−1.96781
$596$	−17.3205	−0.709476
$597$	0	0
$598$	−24.0000	−0.981433
$599$	27.7128	1.13231	0.566157	−	0.824297i	$-0.308429\pi$
$599$	27.7128	1.13231	0.566157	+	0.824297i	$0.308429\pi$
$600$	0	0
$601$	−40.0000	−1.63163	−0.815817	−	0.578310i	$-0.803712\pi$
$601$	−40.0000	−1.63163	−0.815817	+	0.578310i	$0.803712\pi$
$602$	55.4256	2.25898
$603$	0	0
$604$	17.0000	0.691720
$605$	55.4256	2.25337
$606$	0	0
$607$	−43.0000	−1.74532	−0.872658	−	0.488332i	$-0.837606\pi$
$607$	−43.0000	−1.74532	−0.872658	+	0.488332i	$0.837606\pi$
$608$	5.19615	0.210732
$609$	0	0
$610$	6.00000	0.242933
$611$	34.6410	1.40143
$612$	0	0
$613$	−22.0000	−0.888572	−0.444286	−	0.895885i	$-0.646543\pi$
$613$	−22.0000	−0.888572	−0.444286	+	0.895885i	$0.646543\pi$
$614$	−8.66025	−0.349499
$615$	0	0
$616$	36.0000	1.45048
$617$	−34.6410	−1.39459	−0.697297	−	0.716782i	$-0.745614\pi$
$617$	−34.6410	−1.39459	−0.697297	+	0.716782i	$0.745614\pi$
$618$	0	0
$619$	−40.0000	−1.60774	−0.803868	−	0.594808i	$-0.797228\pi$
$619$	−40.0000	−1.60774	−0.803868	+	0.594808i	$0.797228\pi$
$620$	−24.2487	−0.973852
$621$	0	0
$622$	−45.0000	−1.80434
$623$	−6.92820	−0.277573
$624$	0	0
$625$	−11.0000	−0.440000
$626$	12.1244	0.484587
$627$	0	0
$628$	5.00000	0.199522
$629$	−34.6410	−1.38123
$630$	0	0
$631$	−34.0000	−1.35352	−0.676759	−	0.736204i	$-0.736616\pi$
$631$	−34.0000	−1.35352	−0.676759	+	0.736204i	$0.736616\pi$
$632$	−1.73205	−0.0688973
$633$	0	0
$634$	−12.0000	−0.476581
$635$	−13.8564	−0.549875
$636$	0	0
$637$	−36.0000	−1.42637
$638$	15.5885	0.617153
$639$	0	0
$640$	−42.0000	−1.66020
$641$	−12.1244	−0.478883	−0.239442	−	0.970911i	$-0.576964\pi$
$641$	−12.1244	−0.478883	−0.239442	+	0.970911i	$0.576964\pi$
$642$	0	0
$643$	−4.00000	−0.157745	−0.0788723	−	0.996885i	$-0.525132\pi$
$643$	−4.00000	−0.157745	−0.0788723	+	0.996885i	$0.525132\pi$
$644$	13.8564	0.546019
$645$	0	0
$646$	−6.00000	−0.236067
$647$	−36.3731	−1.42997	−0.714986	−	0.699138i	$-0.753567\pi$
$647$	−36.3731	−1.42997	−0.714986	+	0.699138i	$0.753567\pi$
$648$	0	0
$649$	−18.0000	−0.706562
$650$	48.4974	1.90223
$651$	0	0
$652$	−10.0000	−0.391630
$653$	0	0		−	1.00000i	$-0.5\pi$
$653$	0	0			1.00000i	$0.5\pi$
$654$	0	0
$655$	12.0000	0.468879
$656$	8.66025	0.338126
$657$	0	0
$658$	−60.0000	−2.33904
$659$	6.92820	0.269884	0.134942	−	0.990853i	$-0.456915\pi$
$659$	6.92820	0.269884	0.134942	+	0.990853i	$0.456915\pi$
$660$	0	0
$661$	26.0000	1.01128	0.505641	−	0.862744i	$-0.331256\pi$
$661$	26.0000	1.01128	0.505641	+	0.862744i	$0.331256\pi$
$662$	6.92820	0.269272
$663$	0	0
$664$	9.00000	0.349268
$665$	−13.8564	−0.537328
$666$	0	0
$667$	−6.00000	−0.232321
$668$	6.92820	0.268060
$669$	0	0
$670$	78.0000	3.01340
$671$	5.19615	0.200595
$672$	0	0
$673$	−22.0000	−0.848038	−0.424019	−	0.905653i	$-0.639381\pi$
$673$	−22.0000	−0.848038	−0.424019	+	0.905653i	$0.639381\pi$
$674$	−24.2487	−0.934025
$675$	0	0
$676$	3.00000	0.115385
$677$	41.5692	1.59763	0.798817	−	0.601574i	$-0.205459\pi$
$677$	41.5692	1.59763	0.798817	+	0.601574i	$0.205459\pi$
$678$	0	0
$679$	−8.00000	−0.307012
$680$	20.7846	0.797053
$681$	0	0
$682$	−63.0000	−2.41239
$683$	−20.7846	−0.795301	−0.397650	−	0.917537i	$-0.630174\pi$
$683$	−20.7846	−0.795301	−0.397650	+	0.917537i	$0.630174\pi$
$684$	0	0
$685$	60.0000	2.29248
$686$	13.8564	0.529040
$687$	0	0
$688$	−40.0000	−1.52499
$689$	34.6410	1.31972
$690$	0	0
$691$	−10.0000	−0.380418	−0.190209	−	0.981744i	$-0.560917\pi$
$691$	−10.0000	−0.380418	−0.190209	+	0.981744i	$0.560917\pi$
$692$	−12.1244	−0.460899
$693$	0	0
$694$	−6.00000	−0.227757
$695$	48.4974	1.83961
$696$	0	0
$697$	−6.00000	−0.227266
$698$	−39.8372	−1.50786
$699$	0	0
$700$	−28.0000	−1.05830
$701$	−48.4974	−1.83172	−0.915861	−	0.401495i	$-0.868491\pi$
$701$	−48.4974	−1.83172	−0.915861	+	0.401495i	$0.868491\pi$
$702$	0	0
$703$	−10.0000	−0.377157
$704$	−5.19615	−0.195837
$705$	0	0
$706$	54.0000	2.03232
$707$	−69.2820	−2.60562
$708$	0	0
$709$	26.0000	0.976450	0.488225	−	0.872718i	$-0.337644\pi$
$709$	26.0000	0.976450	0.488225	+	0.872718i	$0.337644\pi$
$710$	−20.7846	−0.780033
$711$	0	0
$712$	3.00000	0.112430
$713$	24.2487	0.908121
$714$	0	0
$715$	72.0000	2.69265
$716$	−10.3923	−0.388379
$717$	0	0
$718$	30.0000	1.11959
$719$	−1.73205	−0.0645946	−0.0322973	−	0.999478i	$-0.510282\pi$
$719$	−1.73205	−0.0645946	−0.0322973	+	0.999478i	$0.510282\pi$
$720$	0	0
$721$	64.0000	2.38348
$722$	−1.73205	−0.0644603
$723$	0	0
$724$	14.0000	0.520306
$725$	12.1244	0.450287
$726$	0	0
$727$	26.0000	0.964287	0.482143	−	0.876092i	$-0.339858\pi$
$727$	26.0000	0.964287	0.482143	+	0.876092i	$0.339858\pi$
$728$	27.7128	1.02711
$729$	0	0
$730$	42.0000	1.55449
$731$	27.7128	1.02500
$732$	0	0
$733$	5.00000	0.184679	0.0923396	−	0.995728i	$-0.470565\pi$
$733$	5.00000	0.184679	0.0923396	+	0.995728i	$0.470565\pi$
$734$	48.4974	1.79007
$735$	0	0
$736$	−18.0000	−0.663489
$737$	67.5500	2.48824
$738$	0	0
$739$	2.00000	0.0735712	0.0367856	−	0.999323i	$-0.488288\pi$
$739$	2.00000	0.0735712	0.0367856	+	0.999323i	$0.488288\pi$
$740$	−34.6410	−1.27343
$741$	0	0
$742$	−60.0000	−2.20267
$743$	−38.1051	−1.39794	−0.698971	−	0.715150i	$-0.746358\pi$
$743$	−38.1051	−1.39794	−0.698971	+	0.715150i	$0.746358\pi$
$744$	0	0
$745$	−60.0000	−2.19823
$746$	6.92820	0.253660
$747$	0	0
$748$	−18.0000	−0.658145
$749$	−41.5692	−1.51891
$750$	0	0
$751$	−40.0000	−1.45962	−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000	−1.45962	−0.729810	+	0.683650i	$0.760392\pi$
$752$	43.3013	1.57903
$753$	0	0
$754$	12.0000	0.437014
$755$	58.8897	2.14322
$756$	0	0
$757$	47.0000	1.70824	0.854122	−	0.520073i	$-0.174095\pi$
$757$	47.0000	1.70824	0.854122	+	0.520073i	$0.174095\pi$
$758$	1.73205	0.0629109
$759$	0	0
$760$	6.00000	0.217643
$761$	41.5692	1.50688	0.753442	−	0.657515i	$-0.228392\pi$
$761$	41.5692	1.50688	0.753442	+	0.657515i	$0.228392\pi$
$762$	0	0
$763$	16.0000	0.579239
$764$	15.5885	0.563971
$765$	0	0
$766$	−60.0000	−2.16789
$767$	−13.8564	−0.500326
$768$	0	0
$769$	−22.0000	−0.793340	−0.396670	−	0.917961i	$-0.629834\pi$
$769$	−22.0000	−0.793340	−0.396670	+	0.917961i	$0.629834\pi$
$770$	−124.708	−4.49415
$771$	0	0
$772$	−4.00000	−0.143963
$773$	−19.0526	−0.685273	−0.342636	−	0.939468i	$-0.611320\pi$
$773$	−19.0526	−0.685273	−0.342636	+	0.939468i	$0.611320\pi$
$774$	0	0
$775$	−49.0000	−1.76013
$776$	3.46410	0.124354
$777$	0	0
$778$	−12.0000	−0.430221
$779$	−1.73205	−0.0620572
$780$	0	0
$781$	−18.0000	−0.644091
$782$	20.7846	0.743256
$783$	0	0
$784$	−45.0000	−1.60714
$785$	17.3205	0.618195
$786$	0	0
$787$	−25.0000	−0.891154	−0.445577	−	0.895244i	$-0.647001\pi$
$787$	−25.0000	−0.891154	−0.445577	+	0.895244i	$0.647001\pi$
$788$	0	0
$789$	0	0
$790$	6.00000	0.213470
$791$	20.7846	0.739016
$792$	0	0
$793$	4.00000	0.142044
$794$	−19.0526	−0.676150
$795$	0	0
$796$	20.0000	0.708881
$797$	15.5885	0.552171	0.276086	−	0.961133i	$-0.410963\pi$
$797$	15.5885	0.552171	0.276086	+	0.961133i	$0.410963\pi$
$798$	0	0
$799$	−30.0000	−1.06132
$800$	36.3731	1.28598
$801$	0	0
$802$	−24.0000	−0.847469
$803$	36.3731	1.28358
$804$	0	0
$805$	48.0000	1.69178
$806$	−48.4974	−1.70825
$807$	0	0
$808$	30.0000	1.05540
$809$	−20.7846	−0.730748	−0.365374	−	0.930861i	$-0.619059\pi$
$809$	−20.7846	−0.730748	−0.365374	+	0.930861i	$0.619059\pi$
$810$	0	0
$811$	23.0000	0.807639	0.403820	−	0.914839i	$-0.367682\pi$
$811$	23.0000	0.807639	0.403820	+	0.914839i	$0.367682\pi$
$812$	−6.92820	−0.243132
$813$	0	0
$814$	−90.0000	−3.15450
$815$	−34.6410	−1.21342
$816$	0	0
$817$	8.00000	0.279885
$818$	17.3205	0.605597
$819$	0	0
$820$	−6.00000	−0.209529
$821$	38.1051	1.32988	0.664939	−	0.746898i	$-0.268458\pi$
$821$	38.1051	1.32988	0.664939	+	0.746898i	$0.268458\pi$
$822$	0	0
$823$	26.0000	0.906303	0.453152	−	0.891434i	$-0.350300\pi$
$823$	26.0000	0.906303	0.453152	+	0.891434i	$0.350300\pi$
$824$	−27.7128	−0.965422
$825$	0	0
$826$	24.0000	0.835067
$827$	−13.8564	−0.481834	−0.240917	−	0.970546i	$-0.577448\pi$
$827$	−13.8564	−0.481834	−0.240917	+	0.970546i	$0.577448\pi$
$828$	0	0
$829$	8.00000	0.277851	0.138926	−	0.990303i	$-0.455635\pi$
$829$	8.00000	0.277851	0.138926	+	0.990303i	$0.455635\pi$
$830$	−31.1769	−1.08217
$831$	0	0
$832$	−4.00000	−0.138675
$833$	31.1769	1.08022
$834$	0	0
$835$	24.0000	0.830554
$836$	−5.19615	−0.179713
$837$	0	0
$838$	−27.0000	−0.932700
$839$	−6.92820	−0.239188	−0.119594	−	0.992823i	$-0.538159\pi$
$839$	−6.92820	−0.239188	−0.119594	+	0.992823i	$0.538159\pi$
$840$	0	0
$841$	−26.0000	−0.896552
$842$	−34.6410	−1.19381
$843$	0	0
$844$	−13.0000	−0.447478
$845$	10.3923	0.357506
$846$	0	0
$847$	−64.0000	−2.19907
$848$	43.3013	1.48697
$849$	0	0
$850$	−42.0000	−1.44059
$851$	34.6410	1.18748
$852$	0	0
$853$	17.0000	0.582069	0.291034	−	0.956713i	$-0.406001\pi$
$853$	17.0000	0.582069	0.291034	+	0.956713i	$0.406001\pi$
$854$	−6.92820	−0.237078
$855$	0	0
$856$	18.0000	0.615227
$857$	8.66025	0.295829	0.147914	−	0.989000i	$-0.452744\pi$
$857$	8.66025	0.295829	0.147914	+	0.989000i	$0.452744\pi$
$858$	0	0
$859$	14.0000	0.477674	0.238837	−	0.971060i	$-0.423234\pi$
$859$	14.0000	0.477674	0.238837	+	0.971060i	$0.423234\pi$
$860$	27.7128	0.944999
$861$	0	0
$862$	60.0000	2.04361
$863$	31.1769	1.06127	0.530637	−	0.847599i	$-0.321953\pi$
$863$	31.1769	1.06127	0.530637	+	0.847599i	$0.321953\pi$
$864$	0	0
$865$	−42.0000	−1.42804
$866$	−34.6410	−1.17715
$867$	0	0
$868$	28.0000	0.950382
$869$	5.19615	0.176267
$870$	0	0
$871$	52.0000	1.76195
$872$	−6.92820	−0.234619
$873$	0	0
$874$	6.00000	0.202953
$875$	−27.7128	−0.936864
$876$	0	0
$877$	−40.0000	−1.35070	−0.675352	−	0.737496i	$-0.736008\pi$
$877$	−40.0000	−1.35070	−0.675352	+	0.737496i	$0.736008\pi$
$878$	32.9090	1.11062
$879$	0	0
$880$	90.0000	3.03390
$881$	31.1769	1.05038	0.525188	−	0.850986i	$-0.323995\pi$
$881$	31.1769	1.05038	0.525188	+	0.850986i	$0.323995\pi$
$882$	0	0
$883$	−34.0000	−1.14419	−0.572096	−	0.820187i	$-0.693869\pi$
$883$	−34.0000	−1.14419	−0.572096	+	0.820187i	$0.693869\pi$
$884$	−13.8564	−0.466041
$885$	0	0
$886$	69.0000	2.31810
$887$	27.7128	0.930505	0.465253	−	0.885178i	$-0.345963\pi$
$887$	27.7128	0.930505	0.465253	+	0.885178i	$0.345963\pi$
$888$	0	0
$889$	16.0000	0.536623
$890$	−10.3923	−0.348351
$891$	0	0
$892$	−1.00000	−0.0334825
$893$	−8.66025	−0.289804
$894$	0	0
$895$	−36.0000	−1.20335
$896$	48.4974	1.62019
$897$	0	0
$898$	−63.0000	−2.10234
$899$	−12.1244	−0.404370
$900$	0	0
$901$	−30.0000	−0.999445
$902$	−15.5885	−0.519039
$903$	0	0
$904$	−9.00000	−0.299336
$905$	48.4974	1.61211
$906$	0	0
$907$	20.0000	0.664089	0.332045	−	0.943264i	$-0.392262\pi$
$907$	20.0000	0.664089	0.332045	+	0.943264i	$0.392262\pi$
$908$	10.3923	0.344881
$909$	0	0
$910$	−96.0000	−3.18237
$911$	20.7846	0.688625	0.344312	−	0.938855i	$-0.388112\pi$
$911$	20.7846	0.688625	0.344312	+	0.938855i	$0.388112\pi$
$912$	0	0
$913$	−27.0000	−0.893570
$914$	−39.8372	−1.31770
$915$	0	0
$916$	−7.00000	−0.231287
$917$	−13.8564	−0.457579
$918$	0	0
$919$	8.00000	0.263896	0.131948	−	0.991257i	$-0.457877\pi$
$919$	8.00000	0.263896	0.131948	+	0.991257i	$0.457877\pi$
$920$	−20.7846	−0.685248
$921$	0	0
$922$	66.0000	2.17359
$923$	−13.8564	−0.456089
$924$	0	0
$925$	−70.0000	−2.30159
$926$	58.8897	1.93524
$927$	0	0
$928$	9.00000	0.295439
$929$	13.8564	0.454614	0.227307	−	0.973823i	$-0.427008\pi$
$929$	13.8564	0.454614	0.227307	+	0.973823i	$0.427008\pi$
$930$	0	0
$931$	9.00000	0.294963
$932$	3.46410	0.113470
$933$	0	0
$934$	−9.00000	−0.294489
$935$	−62.3538	−2.03919
$936$	0	0
$937$	29.0000	0.947389	0.473694	−	0.880689i	$-0.342920\pi$
$937$	29.0000	0.947389	0.473694	+	0.880689i	$0.342920\pi$
$938$	−90.0666	−2.94078
$939$	0	0
$940$	−30.0000	−0.978492
$941$	−55.4256	−1.80682	−0.903412	−	0.428774i	$-0.858946\pi$
$941$	−55.4256	−1.80682	−0.903412	+	0.428774i	$0.858946\pi$
$942$	0	0
$943$	6.00000	0.195387
$944$	−17.3205	−0.563735
$945$	0	0
$946$	72.0000	2.34092
$947$	50.2295	1.63224	0.816119	−	0.577883i	$-0.196121\pi$
$947$	50.2295	1.63224	0.816119	+	0.577883i	$0.196121\pi$
$948$	0	0
$949$	28.0000	0.908918
$950$	−12.1244	−0.393366
$951$	0	0
$952$	−24.0000	−0.777844
$953$	29.4449	0.953813	0.476906	−	0.878954i	$-0.341758\pi$
$953$	29.4449	0.953813	0.476906	+	0.878954i	$0.341758\pi$
$954$	0	0
$955$	54.0000	1.74740
$956$	15.5885	0.504167
$957$	0	0
$958$	6.00000	0.193851
$959$	−69.2820	−2.23723
$960$	0	0
$961$	18.0000	0.580645
$962$	−69.2820	−2.23374
$963$	0	0
$964$	−10.0000	−0.322078
$965$	−13.8564	−0.446054
$966$	0	0
$967$	50.0000	1.60789	0.803946	−	0.594703i	$-0.202730\pi$
$967$	50.0000	1.60789	0.803946	+	0.594703i	$0.202730\pi$
$968$	27.7128	0.890724
$969$	0	0
$970$	−12.0000	−0.385297
$971$	−27.7128	−0.889346	−0.444673	−	0.895693i	$-0.646680\pi$
$971$	−27.7128	−0.889346	−0.444673	+	0.895693i	$0.646680\pi$
$972$	0	0
$973$	−56.0000	−1.79528
$974$	74.4782	2.38643
$975$	0	0
$976$	5.00000	0.160046
$977$	25.9808	0.831198	0.415599	−	0.909548i	$-0.363572\pi$
$977$	25.9808	0.831198	0.415599	+	0.909548i	$0.363572\pi$
$978$	0	0
$979$	−9.00000	−0.287641
$980$	31.1769	0.995910
$981$	0	0
$982$	21.0000	0.670137
$983$	13.8564	0.441951	0.220975	−	0.975279i	$-0.429076\pi$
$983$	13.8564	0.441951	0.220975	+	0.975279i	$0.429076\pi$
$984$	0	0
$985$	0	0
$986$	−10.3923	−0.330958
$987$	0	0
$988$	−4.00000	−0.127257
$989$	−27.7128	−0.881216
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	−36.3731	−1.15485
$993$	0	0
$994$	24.0000	0.761234
$995$	69.2820	2.19639
$996$	0	0
$997$	−10.0000	−0.316703	−0.158352	−	0.987383i	$-0.550618\pi$
$997$	−10.0000	−0.316703	−0.158352	+	0.987383i	$0.550618\pi$
$998$	6.92820	0.219308
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	513.2.a.c.1.1	✓	2
3.2	odd	2	inner	513.2.a.c.1.2	yes	2
4.3	odd	2		8208.2.a.bb.1.2		2
12.11	even	2		8208.2.a.bb.1.1		2
19.18	odd	2		9747.2.a.p.1.2		2
57.56	even	2		9747.2.a.p.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
513.2.a.c.1.1	✓	2	1.1	even	1	trivial
513.2.a.c.1.2	yes	2	3.2	odd	2	inner
8208.2.a.bb.1.1		2	12.11	even	2
8208.2.a.bb.1.2		2	4.3	odd	2
9747.2.a.p.1.1		2	57.56	even	2
9747.2.a.p.1.2		2	19.18	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 513 = 3^{3} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	513.a (trivial)

\(f(q)\)	\(=\)	\(q-1.73205 q^{2} +1.00000 q^{4} +3.46410 q^{5} -4.00000 q^{7} +1.73205 q^{8} -6.00000 q^{10} -5.19615 q^{11} -4.00000 q^{13} +6.92820 q^{14} -5.00000 q^{16} +3.46410 q^{17} +1.00000 q^{19} +3.46410 q^{20} +9.00000 q^{22} -3.46410 q^{23} +7.00000 q^{25} +6.92820 q^{26} -4.00000 q^{28} +1.73205 q^{29} -7.00000 q^{31} +5.19615 q^{32} -6.00000 q^{34} -13.8564 q^{35} -10.0000 q^{37} -1.73205 q^{38} +6.00000 q^{40} -1.73205 q^{41} +8.00000 q^{43} -5.19615 q^{44} +6.00000 q^{46} -8.66025 q^{47} +9.00000 q^{49} -12.1244 q^{50} -4.00000 q^{52} -8.66025 q^{53} -18.0000 q^{55} -6.92820 q^{56} -3.00000 q^{58} +3.46410 q^{59} -1.00000 q^{61} +12.1244 q^{62} +1.00000 q^{64} -13.8564 q^{65} -13.0000 q^{67} +3.46410 q^{68} +24.0000 q^{70} +3.46410 q^{71} -7.00000 q^{73} +17.3205 q^{74} +1.00000 q^{76} +20.7846 q^{77} -1.00000 q^{79} -17.3205 q^{80} +3.00000 q^{82} +5.19615 q^{83} +12.0000 q^{85} -13.8564 q^{86} -9.00000 q^{88} +1.73205 q^{89} +16.0000 q^{91} -3.46410 q^{92} +15.0000 q^{94} +3.46410 q^{95} +2.00000 q^{97} -15.5885 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} - 8 q^{7} - 12 q^{10} - 8 q^{13} - 10 q^{16} + 2 q^{19} + 18 q^{22} + 14 q^{25} - 8 q^{28} - 14 q^{31} - 12 q^{34} - 20 q^{37} + 12 q^{40} + 16 q^{43} + 12 q^{46} + 18 q^{49} - 8 q^{52} - 36 q^{55}+ \cdots + 4 q^{97}+O(q^{100}) \) 2 * q + 2 * q^4 - 8 * q^7 - 12 * q^10 - 8 * q^13 - 10 * q^16 + 2 * q^19 + 18 * q^22 + 14 * q^25 - 8 * q^28 - 14 * q^31 - 12 * q^34 - 20 * q^37 + 12 * q^40 + 16 * q^43 + 12 * q^46 + 18 * q^49 - 8 * q^52 - 36 * q^55 - 6 * q^58 - 2 * q^61 + 2 * q^64 - 26 * q^67 + 48 * q^70 - 14 * q^73 + 2 * q^76 - 2 * q^79 + 6 * q^82 + 24 * q^85 - 18 * q^88 + 32 * q^91 + 30 * q^94 + 4 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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